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## MAC 2103

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**MAC 2103**Module 10 lnner Product Spaces I**Learning Objectives**Upon completing this module, you should be able to: • Define and find the inner product, norm and distance in a given inner product space. • Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. • Find the orthogonal complement of a subspace of an inner product space. • Find a basis for the orthogonal complement of a subspace of ℜⁿ spanned by a set of row vectors. • Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. • Know the equivalent statements of an n x n matrix A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**General Vector Spaces II**The major topics in this module: Inner Product Spaces, Inner Products, Norm, Distance, Fundamental Matrix Spaces, Rank, and Orthogonality http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.09**Definition of an Inner Product and Orthogonal Vectors**A. Inner Product: Let be any function from into that satisfies the following conditions for any u, v, z in V: and iff Then defines an inner product for all u, v in V. u and v in V are Orthogonal Vectors iff http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Definition of a Norm of a Vector**B. Norm: Let be any function from into that satisfies the following conditions for any u, v in V: Then defines a norm for all u in V. The special norm that is induced by an inner product is http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Definition of the Distance Between Two Vectors**C. Distance: Let be any function from into that satisfies the following conditions for any u, v, w in V: Then defines the distance for all u, v in V. The special distance that is induced by a norm is http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**in ℜⁿ? For u and v in ℜⁿ, we define and Note: These are our usual norm and distance in ℜⁿ. ℜⁿ is an inner product space with the dot product as its inner product. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**in ℜⁿ? (Cont.) Example 1: Let u = (1,2,-1) and v = (2,0,1) in ℜ³. Find the inner product, norms, and distance. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Matrices) in M22? In M22, the vectors are 2 x 2 matrices. Thus, given two matrices U and V in M22, we define Recall: tr(A) = sum of the entries on the main diagonal of A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Matrices) in M22? (Cont.) Example 2: Let be matrices in the vector space M22. Find the inner product, norms and distance for U and V in M22. Recall: tr(A) = sum of the entries on the main diagonal of A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Matrices) in M22? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Polynomials) in P2? Let P2 be the set of all polynomials of degree less than or equal to two. For two polynomials we define http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Polynomials) in P2? (Cont.) Example 3: Find the inner product, norms, and distance for p and q in P2, where http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Functions) in C[a,b]? Let C[a,b] be the set of all continuous functions on [a,b]. For all f and g in C[a,b], we define http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find an Inner Product, Norm, and Distance for Vectors**(Functions) in C[a,b]? (Cont.) Example 4: Let f = f(x) = 1 and g = g(x) = -x. Find the inner product, norms, and distance on C[2,4]. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Cauchy-Schwarz Inequality**Cauchy-Schwarz Inequality: for all f, g in an inner product space V, with the norm induced by the inner product. We evaluate the Cauchy-Schwarz inequality for the previous examples. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Orthogonality for Vectors (Functions) in C[a,b]**Example 5: Let f = f(x) = 1 and g = g(x) = ex. Find the inner product on C[0,2]. Since the inner product is not zero, f and g are not orthogonal functions in C[0,2]. Example 6: Let f = f(x) = 5 and g = g(x) = cos(x). Find the inner product on C[0,π]. Since the inner product is zero, f and g are orthogonal functions in C[0,π]. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Orthogonality for Vectors (Functions) in C[a,b] (Cont.)**Example 7: Let f = f(x) = 4x and g = g(x) = x2. Find the inner product on C[-1,1]. Since the inner product is zero, f and g are orthogonal functions in C[-1,1]. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Inner Product and Orthogonality for Vectors (Functions) in**C[a,b] (Cont.) Example 8: Let f = f(x) = 1 and g = g(x) = sin(2x). Find the inner product on C[-π,π]. Since the inner product is zero, f and g are orthogonal functions in C[-π,π]. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Generalized Theorem of Pythogoras**Generalized Theorem of Pythogoras: If f and g are orthogonal vectors in an inner product space V, then Proof: Let f and g be orthogonal in V, then http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Generalized Theorem of Pythogoras (Cont.)**The Generalized Theorem of Pythogoras We evaluate the theorem for the previous examples. From example 6, we have http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Generalized Theorem of Pythogoras (Cont.)**From Example 7, we have Notice that we have used orthogonality to obtain our results without directly computing the integrals. Next, we will compute the integrals directly to obtain the result. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Generalized Theorem of Pythogoras (Cont.)**Thus, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find the Cosine of the Angle Between Two Vectors in**an Inner Product Space? Example 9: Let ℜ⁵ have the Euclidean inner product. Find the cosine of the angle between u = (2,1,0,-1,5) and v = (1,-1,-2,3,1). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find the Cosine of the Angle Between Two Vectors in**an Inner Product Space? (Cont.) Example 10: Let P2have the previous inner product. Find the cosine of the angle between p and q. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Quick Review on a Basis for a Vector Space and the Dimension**of a Vector Space Let be nonzero vectors in V. Then for , we have that W = span(S) = {all linear combinations of } is a subspace of V. If S contains some linearly dependent vectors, then the dim(W) < n. If S is a linearly independent set of vectors, then S is a basis for the vector space W. W is a proper subspace of V, if dim(W) < dim(V). If dim(W) = dim(V) = n, then S is a basis for V. A non-trivial vector space V always have two subspaces: the trivial vector space, {0} and V. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Quick Review on a Basis for a Vector Space and the Dimension**of a Vector Space So, we need two conditions for the set S to be a basis of a vector space, it must be a linearly independent set and it must span the vector space. The number of basis vectors in S is the dimension of the vector space. For example: dim(ℜ²)=2, dim(ℜ³)=3, dim(ℜ⁵)=5, dim (P2)=3, dim(M22)=4, and dim(C[a,b]) is infinite. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Rank, Row Rank, and Column Rank of a Matrix**The dimensions of the row space of A, column space of A, and nullspace of A are also called the row rank of A, column rank of A, and nullity of A, respectively. The rank of A = the row rank of A = the column rank of A, and is the number of linearly independent columns in the matrix A. The nullity of A = dim(null(A)) and is the number of free variables in the solution space. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Orthogonal Complement of a Subspace W**The orthogonal complement of W, is the set of all vectors in a finite dimensional inner product space V that are orthogonal to every vector in W. Both W and are subspaces of V where dim(W) + dim( ) = dim(V). The only vector common to W and is 0, and http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**The Four Fundamental Matrix Spaces**Let A be an m x n matrix, then: The null(A) and the row(A) are orthogonal complements in ℜⁿ with respect to the Euclidean inner product. The null(AT) and the col(A) are orthogonal complements in ℜᵐ with respect to the Euclidean inner product. Note that row(A) = col(AT) and col(A) = row(AT). The four fundamental matrix spaces are: row(A) = col(AT), col(A) = row(AT), null(A), and null(AT). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find a Basis for the Orthogonal Complement of the**Subspace of ℜⁿ Spanned by the Vectors? Example 11: Find a basis for the orthogonal complement of the subspace of ℜⁿ spanned by the vectors v1, v2, and v3. Let V = span({v1, v2, v3}). Then, V is a subspace of of ℜⁿ if v1, v2, v3∈ ℜⁿ. Let From the last module, we know that the null(A) is the orthogonal complement to the row(A) = col(AT), and that the null(A) is the solution space of the homogeneous system, Ax = 0. Let v1, v2, v3 be the row vectors of the matrix A. Then, row(A) is the orthogonal complement to null(A) which is the solution space to Ax = 0. We shall use Gauss Elimination to reduce A to a row echelon form to find the basis for null(A). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find a Basis for the Orthogonal Complement of the**Subspace of ℜⁿ Spanned by the Vectors? (Cont.) Since the null(A) is the solution space of the homogeneous system, Ax = 0, the general solution of the system is: Note that column 1 and column 2, with the red leading 1’s, are linearly independent. The corresponding columns of A form a basis for the col(A). Then, dim(col(A)) = 2. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find a Basis for the Orthogonal Complement of the**Subspace of ℜⁿ Spanned by the Vectors? (Cont.) The nonzero rows with the leading 1’s are linearly independent. These rows form a basis for row(A). Then, dim(row(A)) = 2. Note that the row space and column space are both two dimensional, so rank(A) = 2. The rank(A) is the common dimension of the row space of A and the column space of A. So, rank(A) = rowrank(A) = colrank(A) = dim(col(A)) =dim(row(A)) is always true for any m x n matrix A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find a Basis for the Orthogonal Complement of the**Subspace of ℜⁿ Spanned by the Vectors? (Cont.) Thus, the solution vector is: and the vector {w1} forms a basis for the null(A). In other words, {w1} = {(16,19,1)} forms a basis for the orthogonal complement of row(A). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**How to Find a Basis for the Orthogonal Complement of the**Subspace of ℜⁿ Spanned by the Vectors? (Cont.) Since B = {w1} is a basis for null(A), span(B) = null(A), and dim(null(A)) = 1 is the nullity of A. In this example, rank(A) is 2 and the nullity of A is 1, and the number of columns of A is 3. For any m x n matrix A, rank(A) + nullity(A) is n, which is the number of columns of A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Let’s Review Some Equivalent Statements**If A is an n x n matrix, and if TA: ℜⁿ→ ℜⁿ is multiplication by A, than the following are equivalent. (See Theorem 6.2.7) • A is invertible. • Ax = 0 has only the trivial solution • The reduced row-echelon form of A is In. • A is expressible as a product of elementary matrices. • Ax = b is consistent for every n x 1 matrix b. • Ax = b has exactly one solution for every n x 1 matrix b. • det(A) ≠ 0. • The range of TA is ℜⁿ. • TA is one-to-one. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Let’s Review Some Equivalent Statements (Cont.)**• The column vectors of A are linearly independent. • The row vectors of A are linearly independent. • The column vectors of A span ℜⁿ. • The row vectors of A span ℜⁿ. • The column vectors of A form a basis for ℜⁿ. • The row vectors of A form a basis for ℜⁿ. • A has rank n = row rank n = column rank n = dim(col(A)). • A has nullity 0 = dim(null(A)). • The orthogonal complement of the nullspace of A, null(A), is ℜⁿ. • The orthogonal complement of the row space of A, row(A), is {0}. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**What have we learned?**We have learned to: • Define and find the inner product, norm and distance in a given inner product space. • Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. • Find the orthogonal complement of a subspace of an inner product space. • Find a basis for the orthogonal complement of a subspace of ℜⁿ spanned by a set of row vectors. • Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. • Know the equivalent statements of an n x n matrix A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09**Credit**Some of these slides have been adapted/modified in part/whole from the following textbook: • Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09